On the basis property of a certain part of eigenvectors and joined vectors of a self-adjoint operator pencil.

*(Russian)*Zbl 0632.47015The main objects to deal with in the paper under review are the selfadjoint quadratic bundles (or pencils after some authors) of the form \(L(\lambda)=A+\lambda +\lambda^ 2B\), where A and B are selfadjoint compact operators acting on the Hilbert space \({\mathcal H}\) and \(\lambda\in {\mathbb{R}}\). A first result asserts that the span of the eigenvectors (i.e. the solutions y of the equation \(L(\lambda)y=0)\) corresponding to a range of the parameter \(\lambda\in [a,b]\in {\mathbb{R}}\), where \(L'(\lambda)\geq \delta >0\), form a Riesz basis of a vector subspace of finite codimension in \({\mathcal H}\). Then the authors drop the condition on the derivative of L and prove the density of some spaces of eigenvectors (the so called first and secondary systems of eigenspaces) by using the recent local classification results of A. G. Kostyuchenko and A. A. Shkalikov [Funkts. Anal. Prilozh. 17, 38-61 (1983; Zbl 0531.47017)].

In the third part of the paper is proved the existence of a factorization like \(L(\lambda)=(I+BZ+\lambda B)(\lambda -Z)\) for a suitable operator Z and under the assumption that \(<L'(\lambda)y,y>\) is either positive or negative definite for any eigenvector \(y: L(\lambda)y=0.\)

The last (and the main) part of the paper contains the proof of the basis property (in the sense of completeness) of the first and respectively secondary systems of eigenspaces of the bundle L. This simple statement is definitive and completes some old partial known facts due to M. G. Krein and H. Langer.

The present memoir by A. S. Markus and V. I. Matsaev is an important contribution to the modern theory of eigenfunction expansions. The text is well and carefully written.

In the third part of the paper is proved the existence of a factorization like \(L(\lambda)=(I+BZ+\lambda B)(\lambda -Z)\) for a suitable operator Z and under the assumption that \(<L'(\lambda)y,y>\) is either positive or negative definite for any eigenvector \(y: L(\lambda)y=0.\)

The last (and the main) part of the paper contains the proof of the basis property (in the sense of completeness) of the first and respectively secondary systems of eigenspaces of the bundle L. This simple statement is definitive and completes some old partial known facts due to M. G. Krein and H. Langer.

The present memoir by A. S. Markus and V. I. Matsaev is an important contribution to the modern theory of eigenfunction expansions. The text is well and carefully written.

Reviewer: M.Putinar

##### MSC:

47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |

47A70 | (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces |

47A68 | Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators |